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Theorem riotaprop 5753
Description: Properties of a restricted definite description operator. Todo (df-riota 5730 update): can some uses of riota2f 5751 be shortened with this? (Contributed by NM, 23-Nov-2013.)
Hypotheses
Ref Expression
riotaprop.0  |-  F/ x ps
riotaprop.1  |-  B  =  ( iota_ x  e.  A  ph )
riotaprop.2  |-  ( x  =  B  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
riotaprop  |-  ( E! x  e.  A  ph  ->  ( B  e.  A  /\  ps ) )
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    ps( x)    B( x)

Proof of Theorem riotaprop
StepHypRef Expression
1 riotaprop.1 . . 3  |-  B  =  ( iota_ x  e.  A  ph )
2 riotacl 5744 . . 3  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A  ph )  e.  A )
31, 2eqeltrid 2226 . 2  |-  ( E! x  e.  A  ph  ->  B  e.  A )
41eqcomi 2143 . . . 4  |-  ( iota_ x  e.  A  ph )  =  B
5 nfriota1 5737 . . . . . 6  |-  F/_ x
( iota_ x  e.  A  ph )
61, 5nfcxfr 2278 . . . . 5  |-  F/_ x B
7 riotaprop.0 . . . . 5  |-  F/ x ps
8 riotaprop.2 . . . . 5  |-  ( x  =  B  ->  ( ph 
<->  ps ) )
96, 7, 8riota2f 5751 . . . 4  |-  ( ( B  e.  A  /\  E! x  e.  A  ph )  ->  ( ps  <->  (
iota_ x  e.  A  ph )  =  B ) )
104, 9mpbiri 167 . . 3  |-  ( ( B  e.  A  /\  E! x  e.  A  ph )  ->  ps )
113, 10mpancom 418 . 2  |-  ( E! x  e.  A  ph  ->  ps )
123, 11jca 304 1  |-  ( E! x  e.  A  ph  ->  ( B  e.  A  /\  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1331   F/wnf 1436    e. wcel 1480   E!wreu 2418   iota_crio 5729
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-sn 3533  df-pr 3534  df-uni 3737  df-iota 5088  df-riota 5730
This theorem is referenced by:  lble  8712
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